Motivated by the question of the relative complexities of the Graph Isomorphism and the Graph Automorphism problems, we define and study the modular graph automorphism problems. These are the decision problems mod_k-GA which consist, for each k>1, of deciding whether the number of automorphisms of a graph is divisible by k. The mod_k-GA problems all turn out to be intermediate in difficulty between Graph Automorphism and Graph Isomorphism. We define an appropriate search version of mod_k-GA and design an algorithm that polynomial-time reduces the mod_k-GA search problem to the decision problem. Combining this algorithm with an IP protocol, we obtain a randomized polynomial-time checker for mod_k-GA, for all k>1.